Optimal. Leaf size=67 \[ \frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{4}{3};1,\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 a c \sqrt{c+d x^3}} \]
[Out]
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Rubi [A] time = 0.194262, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{x^4 \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{4}{3};1,\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 a c \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 28.5982, size = 53, normalized size = 0.79 \[ \frac{x^{4} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{4}{3},1,\frac{3}{2},\frac{7}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )}}{4 a c^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**3+a)/(d*x**3+c)**(3/2),x)
[Out]
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Mathematica [B] time = 0.363293, size = 332, normalized size = 4.96 \[ \frac{x \left (-\frac{32 a^2 c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (2 b c F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}+\frac{7 a b c x^3 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (2 b c F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-14 a c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}-4\right )}{6 \sqrt{c+d x^3} (a d-b c)} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^3/((a + b*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
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Maple [C] time = 0.05, size = 1069, normalized size = 16. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^3+a)/(d*x^3+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^3 + a)*(d*x^3 + c)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^3 + a)*(d*x^3 + c)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b x^{3}\right ) \left (c + d x^{3}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**3+a)/(d*x**3+c)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^3 + a)*(d*x^3 + c)^(3/2)),x, algorithm="giac")
[Out]